---
title: "Cylindrical points to cartesian vector"
subject: "Electromagnetism"
chapter: 1
key: "semester_3/electromagnetism:cylindrical-vector-to-cartesian-components"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/cylindrical-vector-to-cartesian-components
site: BCT Engineering
---

# Cylindrical points to cartesian vector

Electromagnetism · Chapter 1: Introduction

- Asked in: 2075 Asoj, 3+2 marks
- Video: [Vector Conversion: Cylindrical To Cartesian Coordinates | Electromagnetics | GATE IE](https://www.youtube.com/watch?v=cnRbgFFfOBg) — Ekeeda GATE & ESE
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/cylindrical-vector-to-cartesian-components
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-1

## Question

Express in cartesian components: (a) the vector at A(ρ = 4, φ = 40°, z = −2) that extends to B(ρ = 5, φ = −110°, z = 2); (b) a unit vector at B directed toward A.

## Short answer

Convert A(4,40°,−2) and B(5,−110°,2) to Cartesian: A(3.064,2.571,−2), B(−1.710,−4.698,2). (a) R_AB = −4.77 a_x − 7.27 a_y + 4 a_z. (b) Unit vector from B to A: 0.499 a_x + 0.759 a_y − 0.418 a_z.

## Answer

**Given:** A(ρ = 4, φ = 40°, z = −2), B(ρ = 5, φ = −110°, z = 2).

#### Step 1: Cartesian coordinates of the points

```text
A: x = 4 cos40° = 3.064,  y = 4 sin40° = 2.571,  z = −2
B: x = 5 cos(−110°) = −1.710,  y = 5 sin(−110°) = −4.698,  z = 2
```

#### (a) Vector from A to B

```text
R_AB = B − A = (−1.710 − 3.064) a_x + (−4.698 − 2.571) a_y + (2 + 2) a_z
     = −4.774 a_x − 7.270 a_y + 4 a_z
```

Its magnitude is |R_AB| = √(22.79 + 52.84 + 16) = √91.63 = 9.573.

#### (b) Unit vector at B directed toward A

The direction from B to A is R_BA = A − B = 4.774 a_x + 7.270 a_y − 4 a_z, so

```text
a_BA = R_BA / |R_BA| = (4.774 a_x + 7.270 a_y − 4 a_z) / 9.573
     = 0.499 a_x + 0.759 a_y − 0.418 a_z
```

**Check:** 0.499² + 0.759² + 0.418² = 0.249 + 0.576 + 0.175 = 1.000 ✓

**Answer:**
- (a) R_AB = −4.77 a_x − 7.27 a_y + 4 a_z
- (b) a_BA = 0.499 a_x + 0.759 a_y − 0.418 a_z

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is the first step to find the vector from A to B given in cylindrical coordinates?
   A. Subtract the rho values
   B. Convert both points to Cartesian coordinates
   C. Add the phi angles
   D. Take the cross product of the points
   Answer: B. The points are first converted with x = rho cosφ, y = rho sinφ, z = z, so that the vector B - A can be formed. (Key point: Converting points)

2. What is the vector from A to B in Cartesian components?
   A. -4.77 a_x - 7.27 a_y + 4 a_z
   B. 4.77 a_x + 7.27 a_y - 4 a_z
   C. -4.77 a_x + 7.27 a_y + 4 a_z
   D. -1.71 a_x - 4.70 a_y + 2 a_z
   Answer: A. B - A = (-1.710-3.064, -4.698-2.571, 2+2) = (-4.774, -7.270, 4). (Key point: Vector A to B)

3. How is the unit vector at B directed toward A found?
   A. Divide B - A by its magnitude
   B. Divide A - B by its magnitude 9.573
   C. Use A/|A|
   D. Negate the cylindrical phi angle
   Answer: B. The direction from B to A is R_BA = A - B, the negative of R_AB, divided by the same magnitude 9.573. (Key point: Unit vector toward A from B)

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